Revision: 12th Std >> Mechanical Properties of Fluids MAH-MHT CET (PCM/PCB) Mechanical Properties of Fluids

A state of matter that does not have a definite shape or definite volume, occupies the entire space of the container it is in, and is highly compressible is called a gas.

The normal force (or thrust) exerted by a liquid at rest per unit area of the surface in contact with it is called pressure of liquid or hydrostatic pressure.

The measure of a fluid's resistance to flow is called viscosity.

A state of matter that has a definite volume, takes the shape of its container, and is generally considered to be incompressible (volume does not change significantly under pressure) is called a liquid.

A substance which begins to flow when external force is applied on it is called a fluid. (Liquids and gases are fluids.)

The gaseous envelope surrounding the earth is called the earth's atmosphere.

The difference between the absolute pressure and the atmospheric pressure at a point in a liquid is called gauge pressure.

The total pressure exerted by a fluid, which includes both the atmospheric pressure as well as any other additional pressure due to the fluid itself, is called absolute pressure.

The pressure exerted by the atmosphere on the earth's surface is called atmospheric pressure.

The phenomenon in which the liquid pressure at a point is independent of the quantity of liquid and depends only upon the depth of the point below the liquid surface is called hydrostatic paradox.

The force exerted by the air column on unit cross-sectional area at sea level (= 1.01 × 10⁵ Pa = 1.01 bar) is called atmospheric pressure.

High pressure is an area of the atmosphere where the barometric pressure is higher than its surrounding areas. In this case, the wind from the center of high pressure blows towards the surrounding low-pressure areas.

A low-pressure area is an area in the atmosphere where the pressure is lower than its surrounding areas. In this situation, the wind from the surroundings blows towards the center of low pressure.

The pressure exerted by this mercury column is considered as the pressure of magnitude ‘one atmosphere’ (1 atm).

SI unit of pressure is the pascal (Pa) or Nm⁻²
One Pascal: When a force of one newton acts normally on an area of one square metre (1 m²) then the pressure acting on the surface acting on the surface is called one Pascal.

An instrument that utilises Pascal's Law and the height of a mercury column to measure pressure difference, and gives gauge pressure, is called an open tube manometer.

The instruments used to measure pressure are called pressure meters or pressure gauges or vacuum gauges.

An instrument invented by Torricelli in which the level of mercury in a glass tube gives the pressure is called a mercury barometer.

The angle between tangents drawn at the point of contact to the liquid surface and the solid surface inside the liquid is called the angle of contact for a pair of solid and liquid. It is denoted by θ.

Surface tension is defined as the force acting on a unit length of an imaginary line drawn on the free surface of the liquid, the direction of the force being perpendicular to the line so drawn and acting parallel to the surface.

When a liquid is in contact with a solid, the angle between the tangent drawn to the free surface of the liquid and the surface of solid at the point of contact measured inside the liquid is called the angle of contact.

Surface tension is defined as the force per unit length acting at right angles to an imaginary line drawn on the free surface of the liquid. 

The sphere of influence of a molecule is defined as an imaginary sphere with the molecule at its centre and a radius equal to the range of molecular attraction.

An imaginary sphere drawn round a molecule (taken as centre) with a radius equal to the range of molecular attraction is called the sphere of influence of that molecule.

The property of a liquid due to which its free surface tries to have minimum surface area and behaves as if it were under tension somewhat like a stretched elastic membrane is called surface tension.

OR

The force acting along the surface of a liquid per unit length is called surface tension.

A thin film of liquid near its surface having thickness equal to the molecular range of attraction is called surface film.

The work per unit area done by the force that creates a new surface is called surface energy.

OR

The energy required to increase the surface area of a liquid is called surface energy.

The potential energy is greater for molecules at the surface film as compared to molecules well inside the liquid. This extra energy of the molecule on the surface layer of a liquid is called the surface energy of the liquid.

The difference of pressure between the two sides of a liquid surface, which arises in equilibrium because the pressure inside a bubble or drop is greater than outside, is called excess pressure.

When cohesive forces are stronger than adhesive forces (e.g., mercury in glass), the curved liquid surface formed where cos θ is negative and the liquid level is lower is called a convex meniscus.

When adhesive forces are stronger than cohesive forces, the curved liquid surface formed when the liquid is in contact with a solid is called a concave meniscus.

The angle between the surface of the solid and the tangent drawn to the surface of the liquid at the point of contact on the side of liquid is called the angle of contact of that liquid with that solid.

OR

The angle enclosed between the tangents to the liquid surface and the solid surface inside the liquid, both the tangents being drawn at the point of contact of the liquid with the solid, is called the angle of contact.

The difference of pressure between the two sides of a liquid surface (inside and outside a drop or bubble) due to the property of surface tension, which tends to contract the drop or bubble and compress the matter enclosed, is called Excess Pressure.

The rise or fall of level of liquid in a capillary tube is called capillary action or capillarity.

The phenomenon where a liquid in a capillary tube either ascends or descends relative to the surrounding liquid when a tube of very narrow bore is dipped in it is called capillarity.

The vertical height attained by a liquid in a capillary tube at equilibrium, which is independent of the shape of the capillary provided the radius of meniscus remains the same, is called the capillary rise height (h).

A tube with a hole of very small diameter is called a capillary tube or capillary.

The phenomenon in which a liquid rises in a capillary tube when the angle of contact is acute, or falls when the angle of contact is obtuse, due to the interplay of pressure caused by the liquid column and pressure difference due to surface tension, is called capillary ascent (or descent).

The path, straight or curved, the tangent to which at any point gives the direction of the flow of liquid at that point is called a streamline.

That flow of a liquid in which each element of the liquid passing through a point travels along the same path and with the same velocity as the preceding element passes through that point is called streamline flow.

When a liquid moves with a velocity greater than its critical velocity, the motion of the particles of liquid becomes disordered or irregular. Such a flow is called turbulent flow.

If a liquid is flowing over a horizontal surface with a steady flow and moves in the form of layers of different velocities which do not mix with each other, then the flow of liquid is called laminar flow.

The velocity of liquid flow up to which its flow is streamlined and above which its flow becomes turbulent is called critical velocity.

A pure number which determines the nature of flow of liquid through a pipe, defined as the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid, is called Reynold's number.

The viscous force acting per unit area between two layers of liquid moving with unit velocity gradient is called the coefficient of viscosity (η).

The maximum limit of velocity of a body falling in a viscous fluid, at which the net force acting on the body becomes zero, is called terminal velocity.

The constant maximum velocity acquired by a body while falling through a viscous fluid is called terminal velocity.

If \[\lim_{x\to a^{+}}f\left(x\right)\neq\lim_{x\to a^{-}}f\left(x\right),\] then f(x) is said to be non-removable discontinuous.

If \[\lim_{x\to a^{-}}f\left(x\right)=\lim_{x\to a^{+}}f\left(x\right)\neq f\left(a\right),\] then f(x) is said to be removable discontinuous.

The principle which states that for a non-viscous liquid in streamline flow passing through a tube of varying cross-section, the product of the area of cross-section and the velocity of flow remains constant at every point is called the Equation of Continuity.

A function f(x) is said to be continuous at a point x = a, if the following three conditions are satisfied

1. f is defined at every point on an open interval containing a.
2. \[\lim_{x\to a}f\left(x\right)\] exists.
3. \[\lim_{x\to a}f\left(x\right)=f\left(a\right)\].

The resultant of all the forces exerted by a fluid on a body partly or wholly dipped in it, due to hydrostatic pressure, is called upthrust or buoyancy.

P_gauge = P_absolute​ − P_{atmospheric​}

1 atm = 1.01 × 10⁵ Pa = 1.01 bar = 760 torr

Case 1: θ < 90° (Concave Meniscus)

For a convex or concave surface and a liquid drop: ΔP = \[\frac {2T}{R}\] (one liquid surface)

ΔP = \[\frac {2T}{R}\] (only one liquid surface)

ΔP = 0 (no curvature, no excess pressure)

ΔP = \[\frac {4T}{R}\] (two liquid surfaces — inner and outer)

For a drop on a solid surface: cosθ = \[\frac {T_2-T_1}{​T_3}\]​​,

where T₁ = solid-liquid,

T₂ = solid-air, 

T₃ = liquid-air surface tension

Principle: Pressure due to liquid column = Pressure difference due to surface tension

h = \[\frac {2T}{Rdg}\] = \[\frac {2T cos θ}{rdg}\]

where r = radius of capillary tube and θ = angle of contact.

For two different liquids in the same tube:

\[\frac{h_1}{h_2}=\frac{\rho_2T_1}{\rho_1T_2}\]

For the same liquid in tubes of different radii:

h₁​r₁​ = h₂​r₂​

\[R_n=\frac{v_c\rho D}{\eta}\]

where:

• v_c​ = critical velocity of the liquid
• ρ = density of the liquid
• D = diameter of the pipe
• η = coefficient of viscosity of the liquid

v = \[\frac{2}{9}\cdot\frac{r^2(\rho-\sigma)g}{\eta}\]

where:

• v = terminal velocity
• r = radius of the body
• ρ = density of the body
• σ = density of the fluid
• g = acceleration due to gravity
• η = coefficient of viscosity of the fluid

For a non-viscous liquid in streamline flow passing through a tube of varying cross-section:

av = constant

or equivalently:

a ∝ \[\frac {1}{v}\]

where: 

• a = area of cross-section of the tube
• v = velocity of flow of the liquid

"In a liquid at the same level, the pressure will be the same at all points. If not, then due to pressure difference, the liquid cannot be at rest."

Statement: Pascal's Law states that when pressure is applied to a confined (enclosed) fluid, it is transmitted undiminished and equally in all directions throughout the fluid and to the walls of its container.

Mathematical Expression:

Key Points:

• Pressure changes by the same value at every point inside an incompressible, confined liquid.
• Used in hydraulic machines where a small force on a small area produces a large force on a large area.
• Applications: Hydraulic lift, hydraulic brake, hydraulic press, hydraulic jack.

T = `F/L`

where F = Force (N), L = Length (m)

= SI unit of T = `N/m`

Surface tension can also be written as

T = `W/A`

where W = Work (J), A = Area (m²)

= SI unit of T = `J/m^2`

We know

1J=1N×1m

So,

`J/m^2 = (N * m)/m^2 = N/m`

Both units are the same

`1N/m equiv 1J/m^2`

Statement:

"The viscous force Fᵥ acting on a small sphere falling through a viscous medium is directly proportional to the radius of the sphere r, its velocity (v) through the fluid, and the coefficient of viscosity (η) of the fluid."

Formula:

(b) There is no gravitational force acting downwards. However, when the starting velocity is 20 m/s, the viscous force, which is directly proportional to velocity, becomes maximum and tends to accelerate the ball upwards.

\[\text{ When the ball falls under gravity, }\]

\[\text{ neglecting the density of air: } \]

\[\text{ Mass of the sphere = m }\]

\[\text{ Radius = r }\]

\[\text{ Viscous drag coeff . }= \eta\]

\[\text{Terminal velocity is given by}: \]

\[\text{ mg  }= 6\pi\eta r v_T \]

\[ \Rightarrow \frac{6\pi\eta r v_T}{m} = g . . . (1)\]

\[\text{ Now, at terminal velocity, the acceleration of the ball due to the viscous force is given by: } \]

\[a = \frac{6\pi\eta r v_T}{m}\]

\[\text{ Comparing equations (1) and (2), we find that : } \]

\[ \text{ a = g }\]

Thus, we see that the initial acceleration of the ball will be 9.8 ms - 2  .

(c) The velocity of the ball will decrease with time because of the upward viscous drag. As the force of viscosity is directly proportional to the velocity of the ball, the acceleration due to the viscous force will also decrease.

(d) When all the kinetic energy of the ball is radiated as heat due to the viscous force, the ball comes to rest. 

Statement:

"According to this theorem, the total energy (pressure energy, potential energy and kinetic energy) per unit volume or mass of an incompressible and non-viscous fluid in steady flow through a pipe remains constant throughout the flow, provided there is no source or sink of the fluid along the length of the pipe."

Mathematical Form:

For unit volume of liquid flowing through a pipe:

\[P+\rho gh+\frac{1}{2}\rho v^2\] = constant

where:

• P = pressure energy per unit volume
• ρ = density of the fluid
• g = acceleration due to gravity
• h = height of the fluid (potential energy term)
• v = velocity of the fluid (kinetic energy term)

Applications of Bernoulli's Theorem:

• Speed of efflux
• Venturi tube
• Lifting up of aeroplane
• Working of an atomizer
• Blowing off of roofs by stormy wind

"When a body is partly or wholly dipped in a fluid, the fluid exerts a force on the body due to hydrostatic pressure. At any small portion of the surface of the body, the force exerted by the fluid is perpendicular to the surface and is equal to the pressure at that point multiplied by the area. The resultant of all these constant forces is called upthrust or buoyancy."

• Pressure exerted by a liquid column depends on height and density of the liquid column.
• It is independent of the shape of the containing vessel or total mass of the liquid.
• Atmospheric pressure is maximum at the surface of the earth and decreases as we move up into the atmosphere.

• Hydraulic Press — Two cylinders (C & D) filled with liquid; small force applied on piston P₁ (smaller area A₁) is converted into a very large upward force on piston P₂ (larger area A₂), since A₂ > A₁.
• Hydraulic Lift — Works on Pascal's Law to lift or support heavy objects such as cars and trucks using liquid pressure.
• Hydraulic Brakes — Small force on the brake pedal is instantly transmitted equally through brake fluid to all cylinders, producing a large thrust on the wheels to stop the vehicle.

• A highly soluble impurity increases surface tension, while a partially soluble impurity (e.g., detergent) decreases it; a waterproofing agent increases it.
• Surface tension decreases with increase in temperature, given by T = T₀(1 − αθ), where T₀​ is surface tension at 0°C and α is the temperature coefficient.
• When a soap bubble is charged (positively or negatively), force acts outward on the surface, increasing its radius — thus electrification always decreases surface tension.

• Surface tension depends only on the nature of liquid and is independent of area of surface or length of line considered.
• Surface tension of a liquid decreases with rise of temperature; it is zero at boiling point and vanishes at critical temperature.
• Due to surface tension, a drop or bubble tends to contract, which increases internal pressure — this difference between inside and outside pressure is called excess pressure.
• For a drop and bubble in liquid: Δp = \[\frac {2T}{R}\]​; for a bubble in air: Δp = \[\frac {4T}{R}\]​(two free surfaces).

• When T₂ > T₁, cos θ is +ve, angle is acute — liquid partially wets the solid. (e.g., Kerosene on glass)
• When T₂ < T₁, cos θ is −ve, angle is obtuse — liquid does not wet the solid. (e.g., Mercury on glass)
• When T₂ − T₁ = T₃, cos θ = 1, θ = 0° — liquid completely wets and spreads over the solid. (e.g., Pure water on clean glass)
• When T₂ − T₁ > T₃, cos θ > 1, which is impossible — no drop forms, liquid simply spreads; equilibrium is not possible.

Air flows faster over the curved upper surface of the wing (v large, P small) and slower below (v small, P large). This pressure difference generates an upward lift force on the wing.